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<li><a class="reference internal" href="#"><code class="xref py py-mod docutils literal notranslate"><span class="pre">sklearn.cluster</span></code>.SpectralClustering</a><ul>
<li><a class="reference internal" href="#examples-using-sklearn-cluster-spectralclustering">Examples using <code class="docutils literal notranslate"><span class="pre">sklearn.cluster.SpectralClustering</span></code></a></li>
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  <div class="section" id="sklearn-cluster-spectralclustering">
<h1><a class="reference internal" href="../classes.html#module-sklearn.cluster" title="sklearn.cluster"><code class="xref py py-mod docutils literal notranslate"><span class="pre">sklearn.cluster</span></code></a>.SpectralClustering<a class="headerlink" href="#sklearn-cluster-spectralclustering" title="Permalink to this headline">¶</a></h1>
<dl class="class">
<dt id="sklearn.cluster.SpectralClustering">
<em class="property">class </em><code class="sig-prename descclassname">sklearn.cluster.</code><code class="sig-name descname">SpectralClustering</code><span class="sig-paren">(</span><em class="sig-param">n_clusters=8</em>, <em class="sig-param">eigen_solver=None</em>, <em class="sig-param">n_components=None</em>, <em class="sig-param">random_state=None</em>, <em class="sig-param">n_init=10</em>, <em class="sig-param">gamma=1.0</em>, <em class="sig-param">affinity='rbf'</em>, <em class="sig-param">n_neighbors=10</em>, <em class="sig-param">eigen_tol=0.0</em>, <em class="sig-param">assign_labels='kmeans'</em>, <em class="sig-param">degree=3</em>, <em class="sig-param">coef0=1</em>, <em class="sig-param">kernel_params=None</em>, <em class="sig-param">n_jobs=None</em><span class="sig-paren">)</span><a class="reference external" href="https://github.com/scikit-learn/scikit-learn/blob/5f3c3f037/sklearn/cluster/_spectral.py#L275"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sklearn.cluster.SpectralClustering" title="Permalink to this definition">¶</a></dt>
<dd><p>Apply clustering to a projection of the normalized Laplacian.</p>
<p>In practice Spectral Clustering is very useful when the structure of
the individual clusters is highly non-convex or more generally when
a measure of the center and spread of the cluster is not a suitable
description of the complete cluster. For instance when clusters are
nested circles on the 2D plane.</p>
<p>If affinity is the adjacency matrix of a graph, this method can be
used to find normalized graph cuts.</p>
<p>When calling <code class="docutils literal notranslate"><span class="pre">fit</span></code>, an affinity matrix is constructed using either
kernel function such the Gaussian (aka RBF) kernel of the euclidean
distanced <code class="docutils literal notranslate"><span class="pre">d(X,</span> <span class="pre">X)</span></code>:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">gamma</span> <span class="o">*</span> <span class="n">d</span><span class="p">(</span><span class="n">X</span><span class="p">,</span><span class="n">X</span><span class="p">)</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span>
</pre></div>
</div>
<p>or a k-nearest neighbors connectivity matrix.</p>
<p>Alternatively, using <code class="docutils literal notranslate"><span class="pre">precomputed</span></code>, a user-provided affinity
matrix can be used.</p>
<p>Read more in the <a class="reference internal" href="../clustering.html#spectral-clustering"><span class="std std-ref">User Guide</span></a>.</p>
<dl class="field-list">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><dl>
<dt><strong>n_clusters</strong><span class="classifier">integer, optional</span></dt><dd><p>The dimension of the projection subspace.</p>
</dd>
<dt><strong>eigen_solver</strong><span class="classifier">{None, ‘arpack’, ‘lobpcg’, or ‘amg’}</span></dt><dd><p>The eigenvalue decomposition strategy to use. AMG requires pyamg
to be installed. It can be faster on very large, sparse problems,
but may also lead to instabilities.</p>
</dd>
<dt><strong>n_components</strong><span class="classifier">integer, optional, default=n_clusters</span></dt><dd><p>Number of eigen vectors to use for the spectral embedding</p>
</dd>
<dt><strong>random_state</strong><span class="classifier">int, RandomState instance or None (default)</span></dt><dd><p>A pseudo random number generator used for the initialization of the
lobpcg eigen vectors decomposition when <code class="docutils literal notranslate"><span class="pre">eigen_solver='amg'</span></code> and by
the K-Means initialization. Use an int to make the randomness
deterministic.
See <a class="reference internal" href="../../glossary.html#term-random-state"><span class="xref std std-term">Glossary</span></a>.</p>
</dd>
<dt><strong>n_init</strong><span class="classifier">int, optional, default: 10</span></dt><dd><p>Number of time the k-means algorithm will be run with different
centroid seeds. The final results will be the best output of
n_init consecutive runs in terms of inertia.</p>
</dd>
<dt><strong>gamma</strong><span class="classifier">float, default=1.0</span></dt><dd><p>Kernel coefficient for rbf, poly, sigmoid, laplacian and chi2 kernels.
Ignored for <code class="docutils literal notranslate"><span class="pre">affinity='nearest_neighbors'</span></code>.</p>
</dd>
<dt><strong>affinity</strong><span class="classifier">string or callable, default ‘rbf’</span></dt><dd><dl class="simple">
<dt>How to construct the affinity matrix.</dt><dd><ul class="simple">
<li><p>‘nearest_neighbors’ : construct the affinity matrix by computing a
graph of nearest neighbors.</p></li>
<li><p>‘rbf’ : construct the affinity matrix using a radial basis function
(RBF) kernel.</p></li>
<li><p>‘precomputed’ : interpret <code class="docutils literal notranslate"><span class="pre">X</span></code> as a precomputed affinity matrix.</p></li>
<li><p>‘precomputed_nearest_neighbors’ : interpret <code class="docutils literal notranslate"><span class="pre">X</span></code> as a sparse graph
of precomputed nearest neighbors, and constructs the affinity matrix
by selecting the <code class="docutils literal notranslate"><span class="pre">n_neighbors</span></code> nearest neighbors.</p></li>
<li><p>one of the kernels supported by
<code class="xref py py-func docutils literal notranslate"><span class="pre">pairwise_kernels</span></code>.</p></li>
</ul>
</dd>
</dl>
<p>Only kernels that produce similarity scores (non-negative values that
increase with similarity) should be used. This property is not checked
by the clustering algorithm.</p>
</dd>
<dt><strong>n_neighbors</strong><span class="classifier">integer</span></dt><dd><p>Number of neighbors to use when constructing the affinity matrix using
the nearest neighbors method. Ignored for <code class="docutils literal notranslate"><span class="pre">affinity='rbf'</span></code>.</p>
</dd>
<dt><strong>eigen_tol</strong><span class="classifier">float, optional, default: 0.0</span></dt><dd><p>Stopping criterion for eigendecomposition of the Laplacian matrix
when <code class="docutils literal notranslate"><span class="pre">eigen_solver='arpack'</span></code>.</p>
</dd>
<dt><strong>assign_labels</strong><span class="classifier">{‘kmeans’, ‘discretize’}, default: ‘kmeans’</span></dt><dd><p>The strategy to use to assign labels in the embedding
space. There are two ways to assign labels after the laplacian
embedding. k-means can be applied and is a popular choice. But it can
also be sensitive to initialization. Discretization is another approach
which is less sensitive to random initialization.</p>
</dd>
<dt><strong>degree</strong><span class="classifier">float, default=3</span></dt><dd><p>Degree of the polynomial kernel. Ignored by other kernels.</p>
</dd>
<dt><strong>coef0</strong><span class="classifier">float, default=1</span></dt><dd><p>Zero coefficient for polynomial and sigmoid kernels.
Ignored by other kernels.</p>
</dd>
<dt><strong>kernel_params</strong><span class="classifier">dictionary of string to any, optional</span></dt><dd><p>Parameters (keyword arguments) and values for kernel passed as
callable object. Ignored by other kernels.</p>
</dd>
<dt><strong>n_jobs</strong><span class="classifier">int or None, optional (default=None)</span></dt><dd><p>The number of parallel jobs to run.
<code class="docutils literal notranslate"><span class="pre">None</span></code> means 1 unless in a <a class="reference external" href="https://joblib.readthedocs.io/en/latest/parallel.html#joblib.parallel_backend" title="(in joblib v0.14.1.dev0)"><code class="xref py py-obj docutils literal notranslate"><span class="pre">joblib.parallel_backend</span></code></a> context.
<code class="docutils literal notranslate"><span class="pre">-1</span></code> means using all processors. See <a class="reference internal" href="../../glossary.html#term-n-jobs"><span class="xref std std-term">Glossary</span></a>
for more details.</p>
</dd>
</dl>
</dd>
<dt class="field-even">Attributes</dt>
<dd class="field-even"><dl class="simple">
<dt><strong>affinity_matrix_</strong><span class="classifier">array-like, shape (n_samples, n_samples)</span></dt><dd><p>Affinity matrix used for clustering. Available only if after calling
<code class="docutils literal notranslate"><span class="pre">fit</span></code>.</p>
</dd>
<dt><strong>labels_</strong><span class="classifier">array, shape (n_samples,)</span></dt><dd><p>Labels of each point</p>
</dd>
</dl>
</dd>
</dl>
<p class="rubric">Notes</p>
<p>If you have an affinity matrix, such as a distance matrix,
for which 0 means identical elements, and high values means
very dissimilar elements, it can be transformed in a
similarity matrix that is well suited for the algorithm by
applying the Gaussian (RBF, heat) kernel:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span> <span class="n">dist_matrix</span> <span class="o">**</span> <span class="mi">2</span> <span class="o">/</span> <span class="p">(</span><span class="mf">2.</span> <span class="o">*</span> <span class="n">delta</span> <span class="o">**</span> <span class="mi">2</span><span class="p">))</span>
</pre></div>
</div>
<p>Where <code class="docutils literal notranslate"><span class="pre">delta</span></code> is a free parameter representing the width of the Gaussian
kernel.</p>
<p>Another alternative is to take a symmetric version of the k
nearest neighbors connectivity matrix of the points.</p>
<p>If the pyamg package is installed, it is used: this greatly
speeds up computation.</p>
<p class="rubric">References</p>
<ul class="simple">
<li><p>Normalized cuts and image segmentation, 2000
Jianbo Shi, Jitendra Malik
<a class="reference external" href="http://citeseer.ist.psu.edu/viewdoc/summary?doi=10.1.1.160.2324">http://citeseer.ist.psu.edu/viewdoc/summary?doi=10.1.1.160.2324</a></p></li>
<li><p>A Tutorial on Spectral Clustering, 2007
Ulrike von Luxburg
<a class="reference external" href="http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.165.9323">http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.165.9323</a></p></li>
<li><p>Multiclass spectral clustering, 2003
Stella X. Yu, Jianbo Shi
<a class="reference external" href="https://www1.icsi.berkeley.edu/~stellayu/publication/doc/2003kwayICCV.pdf">https://www1.icsi.berkeley.edu/~stellayu/publication/doc/2003kwayICCV.pdf</a></p></li>
</ul>
<p class="rubric">Examples</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sklearn.cluster</span> <span class="kn">import</span> <span class="n">SpectralClustering</span>
<span class="gp">&gt;&gt;&gt; </span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span>
<span class="gp">... </span>              <span class="p">[</span><span class="mi">4</span><span class="p">,</span> <span class="mi">7</span><span class="p">],</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">],</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">6</span><span class="p">]])</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">clustering</span> <span class="o">=</span> <span class="n">SpectralClustering</span><span class="p">(</span><span class="n">n_clusters</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span>
<span class="gp">... </span>        <span class="n">assign_labels</span><span class="o">=</span><span class="s2">&quot;discretize&quot;</span><span class="p">,</span>
<span class="gp">... </span>        <span class="n">random_state</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">clustering</span><span class="o">.</span><span class="n">labels_</span>
<span class="go">array([1, 1, 1, 0, 0, 0])</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">clustering</span>
<span class="go">SpectralClustering(assign_labels=&#39;discretize&#39;, n_clusters=2,</span>
<span class="go">    random_state=0)</span>
</pre></div>
</div>
<p class="rubric">Methods</p>
<table class="longtable docutils align-default">
<colgroup>
<col style="width: 10%" />
<col style="width: 90%" />
</colgroup>
<tbody>
<tr class="row-odd"><td><p><a class="reference internal" href="#sklearn.cluster.SpectralClustering.fit" title="sklearn.cluster.SpectralClustering.fit"><code class="xref py py-obj docutils literal notranslate"><span class="pre">fit</span></code></a>(self, X[, y])</p></td>
<td><p>Perform spectral clustering from features, or affinity matrix.</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="#sklearn.cluster.SpectralClustering.fit_predict" title="sklearn.cluster.SpectralClustering.fit_predict"><code class="xref py py-obj docutils literal notranslate"><span class="pre">fit_predict</span></code></a>(self, X[, y])</p></td>
<td><p>Perform spectral clustering from features, or affinity matrix, and return cluster labels.</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="#sklearn.cluster.SpectralClustering.get_params" title="sklearn.cluster.SpectralClustering.get_params"><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_params</span></code></a>(self[, deep])</p></td>
<td><p>Get parameters for this estimator.</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="#sklearn.cluster.SpectralClustering.set_params" title="sklearn.cluster.SpectralClustering.set_params"><code class="xref py py-obj docutils literal notranslate"><span class="pre">set_params</span></code></a>(self, \*\*params)</p></td>
<td><p>Set the parameters of this estimator.</p></td>
</tr>
</tbody>
</table>
<dl class="method">
<dt id="sklearn.cluster.SpectralClustering.__init__">
<code class="sig-name descname">__init__</code><span class="sig-paren">(</span><em class="sig-param">self</em>, <em class="sig-param">n_clusters=8</em>, <em class="sig-param">eigen_solver=None</em>, <em class="sig-param">n_components=None</em>, <em class="sig-param">random_state=None</em>, <em class="sig-param">n_init=10</em>, <em class="sig-param">gamma=1.0</em>, <em class="sig-param">affinity='rbf'</em>, <em class="sig-param">n_neighbors=10</em>, <em class="sig-param">eigen_tol=0.0</em>, <em class="sig-param">assign_labels='kmeans'</em>, <em class="sig-param">degree=3</em>, <em class="sig-param">coef0=1</em>, <em class="sig-param">kernel_params=None</em>, <em class="sig-param">n_jobs=None</em><span class="sig-paren">)</span><a class="reference external" href="https://github.com/scikit-learn/scikit-learn/blob/5f3c3f037/sklearn/cluster/_spectral.py#L437"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sklearn.cluster.SpectralClustering.__init__" title="Permalink to this definition">¶</a></dt>
<dd><p>Initialize self.  See help(type(self)) for accurate signature.</p>
</dd></dl>

<dl class="method">
<dt id="sklearn.cluster.SpectralClustering.fit">
<code class="sig-name descname">fit</code><span class="sig-paren">(</span><em class="sig-param">self</em>, <em class="sig-param">X</em>, <em class="sig-param">y=None</em><span class="sig-paren">)</span><a class="reference external" href="https://github.com/scikit-learn/scikit-learn/blob/5f3c3f037/sklearn/cluster/_spectral.py#L456"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sklearn.cluster.SpectralClustering.fit" title="Permalink to this definition">¶</a></dt>
<dd><p>Perform spectral clustering from features, or affinity matrix.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><dl class="simple">
<dt><strong>X</strong><span class="classifier">array-like or sparse matrix, shape (n_samples, n_features), or             array-like, shape (n_samples, n_samples)</span></dt><dd><p>Training instances to cluster, or similarities / affinities between
instances if <code class="docutils literal notranslate"><span class="pre">affinity='precomputed'</span></code>. If a sparse matrix is
provided in a format other than <code class="docutils literal notranslate"><span class="pre">csr_matrix</span></code>, <code class="docutils literal notranslate"><span class="pre">csc_matrix</span></code>,
or <code class="docutils literal notranslate"><span class="pre">coo_matrix</span></code>, it will be converted into a sparse
<code class="docutils literal notranslate"><span class="pre">csr_matrix</span></code>.</p>
</dd>
<dt><strong>y</strong><span class="classifier">Ignored</span></dt><dd><p>Not used, present here for API consistency by convention.</p>
</dd>
</dl>
</dd>
<dt class="field-even">Returns</dt>
<dd class="field-even"><dl class="simple">
<dt>self</dt><dd></dd>
</dl>
</dd>
</dl>
</dd></dl>

<dl class="method">
<dt id="sklearn.cluster.SpectralClustering.fit_predict">
<code class="sig-name descname">fit_predict</code><span class="sig-paren">(</span><em class="sig-param">self</em>, <em class="sig-param">X</em>, <em class="sig-param">y=None</em><span class="sig-paren">)</span><a class="reference external" href="https://github.com/scikit-learn/scikit-learn/blob/5f3c3f037/sklearn/cluster/_spectral.py#L523"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sklearn.cluster.SpectralClustering.fit_predict" title="Permalink to this definition">¶</a></dt>
<dd><p>Perform spectral clustering from features, or affinity matrix,
and return cluster labels.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><dl class="simple">
<dt><strong>X</strong><span class="classifier">array-like or sparse matrix, shape (n_samples, n_features), or             array-like, shape (n_samples, n_samples)</span></dt><dd><p>Training instances to cluster, or similarities / affinities between
instances if <code class="docutils literal notranslate"><span class="pre">affinity='precomputed'</span></code>. If a sparse matrix is
provided in a format other than <code class="docutils literal notranslate"><span class="pre">csr_matrix</span></code>, <code class="docutils literal notranslate"><span class="pre">csc_matrix</span></code>,
or <code class="docutils literal notranslate"><span class="pre">coo_matrix</span></code>, it will be converted into a sparse
<code class="docutils literal notranslate"><span class="pre">csr_matrix</span></code>.</p>
</dd>
<dt><strong>y</strong><span class="classifier">Ignored</span></dt><dd><p>Not used, present here for API consistency by convention.</p>
</dd>
</dl>
</dd>
<dt class="field-even">Returns</dt>
<dd class="field-even"><dl class="simple">
<dt><strong>labels</strong><span class="classifier">ndarray, shape (n_samples,)</span></dt><dd><p>Cluster labels.</p>
</dd>
</dl>
</dd>
</dl>
</dd></dl>

<dl class="method">
<dt id="sklearn.cluster.SpectralClustering.get_params">
<code class="sig-name descname">get_params</code><span class="sig-paren">(</span><em class="sig-param">self</em>, <em class="sig-param">deep=True</em><span class="sig-paren">)</span><a class="reference external" href="https://github.com/scikit-learn/scikit-learn/blob/5f3c3f037/sklearn/base.py#L173"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sklearn.cluster.SpectralClustering.get_params" title="Permalink to this definition">¶</a></dt>
<dd><p>Get parameters for this estimator.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><dl class="simple">
<dt><strong>deep</strong><span class="classifier">bool, default=True</span></dt><dd><p>If True, will return the parameters for this estimator and
contained subobjects that are estimators.</p>
</dd>
</dl>
</dd>
<dt class="field-even">Returns</dt>
<dd class="field-even"><dl class="simple">
<dt><strong>params</strong><span class="classifier">mapping of string to any</span></dt><dd><p>Parameter names mapped to their values.</p>
</dd>
</dl>
</dd>
</dl>
</dd></dl>

<dl class="method">
<dt id="sklearn.cluster.SpectralClustering.set_params">
<code class="sig-name descname">set_params</code><span class="sig-paren">(</span><em class="sig-param">self</em>, <em class="sig-param">**params</em><span class="sig-paren">)</span><a class="reference external" href="https://github.com/scikit-learn/scikit-learn/blob/5f3c3f037/sklearn/base.py#L205"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sklearn.cluster.SpectralClustering.set_params" title="Permalink to this definition">¶</a></dt>
<dd><p>Set the parameters of this estimator.</p>
<p>The method works on simple estimators as well as on nested objects
(such as pipelines). The latter have parameters of the form
<code class="docutils literal notranslate"><span class="pre">&lt;component&gt;__&lt;parameter&gt;</span></code> so that it’s possible to update each
component of a nested object.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><dl class="simple">
<dt><strong>**params</strong><span class="classifier">dict</span></dt><dd><p>Estimator parameters.</p>
</dd>
</dl>
</dd>
<dt class="field-even">Returns</dt>
<dd class="field-even"><dl class="simple">
<dt><strong>self</strong><span class="classifier">object</span></dt><dd><p>Estimator instance.</p>
</dd>
</dl>
</dd>
</dl>
</dd></dl>

</dd></dl>

<div class="section" id="examples-using-sklearn-cluster-spectralclustering">
<h2>Examples using <code class="docutils literal notranslate"><span class="pre">sklearn.cluster.SpectralClustering</span></code><a class="headerlink" href="#examples-using-sklearn-cluster-spectralclustering" title="Permalink to this headline">¶</a></h2>
<div class="sphx-glr-thumbcontainer" tooltip="This example shows characteristics of different clustering algorithms on datasets that are &quot;int..."><div class="figure align-default" id="id1">
<img alt="../../_images/sphx_glr_plot_cluster_comparison_thumb.png" src="../../_images/sphx_glr_plot_cluster_comparison_thumb.png" />
<p class="caption"><span class="caption-text"><a class="reference internal" href="../../auto_examples/cluster/plot_cluster_comparison.html#sphx-glr-auto-examples-cluster-plot-cluster-comparison-py"><span class="std std-ref">Comparing different clustering algorithms on toy datasets</span></a></span><a class="headerlink" href="#id1" title="Permalink to this image">¶</a></p>
</div>
</div><div class="clearer"></div></div>
</div>


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